Name: Orthogonal Curves
Uploaded: 2019-07-19
Duration: 11 min 48 s
Description: Have you heard the term 'orthogonal curves'? Can you find angles between two curves? Do watch this video and find all such answers!! Have fun!!

Question

A pair of tangents are drawn to a parabola and are equally inclined to a straight line whose inclination to the axis is

α

; prove that the locus of their point of intersection is the straight line

y = (x - a) \tan 2 α .

Accepted Answer

Let the equation of parabola be $y^{2} = 4 a x$ , then the equation of the tangent in the slope form is:

$y = m x + \frac{a}{m} . . . (1)$

Given, the tangents through $P (h, k)$ are equally inclined to the line whose inclination to the $x -$ axis is $α$ .

Then, if $θ_{1,} θ_{2}$ are angles of inclination of the tangents from $x$ -axis, we have
Then $θ_{2} - α = α - θ_{1}$ (external angle property)
Or $θ_{1} + θ_{2} = 2 α$
Or $\tan (θ_{1} + θ_{2}) = \tan 2 α$
Or $\frac{\tan θ_{1} + \tan θ_{2}}{1 - \tan θ_{1} \tan θ_{2}} = \tan 2 α = \frac{m_{1} + m_{2}}{1 - m_{1} m_{2}}$ , where $m_{1}$ and $m_{2}$ are the slopes of the tangents.

Since, equation $(1)$ is satisfied by the external point $P (h, k)$ , so we get

$k = m h + \frac{a}{m}$ or $m^{2} h - m k + a = 0$ .

Using the relation between the roots and coefficients of a quadratic equation, we have

$m_{1} + m_{2} = \frac{k}{h} . . . (2)$

and $m_{1} m_{2} = \frac{a}{h} . . . (3)$

Putting the values from equation number $(2) and (3)$ , we get
$\frac{(\frac{k}{h})}{(1 - \frac{a}{h})} = \tan 2 α$ or $\frac{k}{h - a} = \tan 2 α$
Generalizing and simplifying the required locus of $(h, k)$ is
$y = (x - a) \tan 2 α$

A pair of tangents are drawn to a parabola and are equally inclined to a straight line whose inclination to the axis is $α$ ; prove that the locus of their point of intersection is the straight line $y = (x - a) \tan 2 α .$

Important Questions on Parabola

Learn and Prepare for any exam you want !

A pair of tangents are drawn to a parabola and are equally inclined to a straight line whose inclination to the axis is α; prove that the locus of their point of intersection is the straight line y=(x-a)tan2α.

Important Questions on Parabola

Learn and Prepare for any exam you want !

A pair of tangents are drawn to a parabola and are equally inclined to a straight line whose inclination to the axis is $α$ ; prove that the locus of their point of intersection is the straight line $y = (x - a) \tan 2 α .$